External EM multiplicative motivic complexes and actual Chern map

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Let $F=\mathbb{Q}$ or ${\mathbb{F}}_p$. The supplied EM complexes $C_b(X)=\mathbb{Z}(b{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)$ on qcqs $F$-schemes are the shifted graded pieces of the specified multiplicative filtration. In the terminology of EM §2.2, the graded spectra $G_b=C_b[2b]$ form a natural commutative algebra object for graded Day convolution: there are coherent associative symmetric unital products $G_b\wedge G_d\to G_{b+d}$ and unit $\mathbb{S}\to G_0$, and pullbacks preserve this data. Equivalently these induce the usual motivic cup products, with even weight shifts. The negative graded pieces vanish, so $C_b=0$ for $b<0$. There is the actual natural map of complex-valued presheaves $c_1:R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}(X,{\mathbb{G}}_m)[-1]⟶C_1(X)$ of Lemma5.1/Definition5.2. Its degree-two source is $\mathrm{Pic}(X)$ with tensor-product group law. It agrees with the classical class on smooth $F$-schemes. The maps denoted $c_1(\mathcal{O}(1){)}^m{\pi }^{\ast }$ in §5 are defined using this class and the multiplication just specified. EM writes $P_X^{\mathrm{l}\mathrm{i}\mathrm{n}}(V)$ for the projectivization classifying rank-one subbundles of $V$, with tautological subbundle $\mathcal{O}(-1)$ and its dual $\mathcal{O}(1)$.

These are the stated construction and conventions at the exact locators in the source reference, restricted to the completed motivic-complex supplier. Definitions4.9/4.23 take pullbacks in filtered E∞-algebras; §2.2 gives the symmetric monoidal associated-graded construction. Definition5.2 supplies the complex map, rather than a class chosen from group agreement. Import those established foundations. Restriction along the prime-field forgetful functor supplies the same interface on qcqs schemes over any field. No replacement of the supplied motivic theory or K-theory model is made.